Of course, the crossover burst size \(\Delta _c\) , which separates the two power law regimes,
depends on the rewiring probability, i.e. \(\Delta _c\)  gradually shifts to smaller values
with increasing \(p\)  in such a way that in the limit \(p\rightarrow 1\)  practically a single power law remains with a significantly lower exponent than that of the original square lattice at \(p=0\) . To characterize this evolution of \(p(\Delta )\)  we determined the average size of the largest avalanche
\(\left<\Delta _{max}\right>\)  and the power law exponent \(\tau \) 
of the regime of large avalanches as function of the rewiring probability \(p\) .
It can be observed in Fig. REF  for the Weibull parameter \(m=1\) 
that up to the rewiring probability \(p_l\approx 0.01\) 
the cutoff cascade size \(\left<\Delta _{max}\right>\)  is nearly constant, although the exponent \(\tau \)  suffered some change.
This behaviour implies that the small fraction of randomized contacts has a minor effect on the cascading
failure dynamics in this parameter range \(p\lesssim 0.01\) . However, above this threshold probability
a rapid change of the cascade size distribution sets on indicated by the steep increase of \(\left<\Delta _{max}\right>\)  and decrease of the exponent \(\tau \) . For high values of the rewiring
probability \(p\rightarrow 1\) , the exponent \(\tau \)  converges to a constant \(\tau \approx 2.3\) ,
which falls very close to the mean field burst size exponent of FBMs \(\tau ^{ELS}=5/2\)  [1]}, [2]}. The result
indicates that on sufficiently randomized load transmission networks the statistics of failure cascades
of the localized load sharing FBM becomes equivalent to the mean field universality class of the system
in agreement with the behaviour of the macroscopic strength of the bundle.
The result is consistent with Ref. [3]} where FBMs were analyzed
on Watts-Strogatz networks in the range of high rewiring probabilities \(p\ge 0.2\)  recovering the mean
field behaviour.
